In Iran's capital city, everyone seems to be living a double life. Leyla, a prostitute, services a top cleric who urges her to pray.

Ever hear the Iranian joke about the mullah riding the bus? The cleric says to the driver: "My friend, since you're on the road half the year, how can you be sure your children are really yours?" The driver is unfazed by the insult: "Excellence, I keep them until age 6. After that, if they look like me, no problem. If not, I ship them off to the seminary."

Subversive humor is a leitmotif of Persian civilization. But over the past 35 years, the Islamic Republic has prohibited bawdy punchlines—along with...